nLab
universe in a topos

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Universes

Foundations

Contents

Idea

A universe in a topos is a topos-theoretic version of the notion of Grothendieck universe; see that page for general motivation and applications.

To free the notion from membership-based set theory, we must replace sets of sets by families of sets, just as in passing from power sets to power objects we must replace sets of subsets by families of subsets.

Definition

A universe in a topos \mathcal{E} is a morphism el:EUel\colon E \to U satisfying the axioms to follow. We think of el:EUel\colon E \to U as a UU-indexed family of objects (sets), and we define a morphism a:AIa\colon A \to I (regarded as an II-indexed family of objects) to be UU-small if there exists a morphism f:IUf\colon I \to U and a pullback square

A E a el I f U.\array{A & \to & E\\ ^a\downarrow && \downarrow^{el}\\ I& \underset{f}{\to} & U.}

Note that ff is not, in general, unique: a universe can contain many isomorphic sets. With this definition, the pullback of a UU-small morphism is automatically again UU-small. We say that an object XX is UU-small if X1X\to 1 is UU-small.

The axioms which must be satisfied are:

  1. Every monomorphism is UU-small.

  2. The composite of UU-small morphisms is UU-small.

  3. If f:AIf\colon A \to I and g:BAg\colon B \to A are UU-small, then so is the dependent product Π fg\Pi_f g (where Π f\Pi_f is the right adjoint to f *:/I/Af^*\colon \mathcal{E}/I \to \mathcal{E}/A).

  4. The subobject classifier Ω\Omega is UU-small.

Note that since 0Ω0\to \Omega is a monomorphism, (1) and (4) imply that the initial object 00 is UU-small. A predicative universe is a morphism el:EUel\colon E \to U where instead of (4) we assume merely that 00 is UU-small; this makes sense in any locally cartesian closed category. In a topos, the generic subobject 1Ω1\to \Omega is a predicative universe, and of course a morphism is Ω\Omega-small if and only if it is a monomorphism.

If we assume only (1)–(3), then the identity morphism 1 0:001_0\colon 0 \to 0 of the initial object would be a universe, for which it itself is the only UU-small morphism. On the other hand, if \mathcal{E} has a natural numbers object NN, we may additionally assume that NN is UU-small, to ensure that all universes contain “infinite” sets.

Note that any object isomorphic to a UU-small object is UU-small; thus in the language of Grothendieck universes this notion of smallness corresponds to essential smallness. Roughly, we may say that (1) corresponds to transitivity of a Grothendieck universe, (3) and (4) correspond to closure under power sets, and (2) corresponds to closure under indexed unions.

Example: universes in SETSET

We spell out in detail some implications of these axioms for the case that the topos in question is the Categeory of Sets according to ETCS, to be denoted SETSET.

Write ** for the terminal object in SETSET, the singleton set. Notice that for each ordinary element uUu \in U, i.e. *uU* \stackrel{u}{\to} U, there is the set E uE_u over uu, defined as the pullback

E u E * u U\array{ E_u &\to& E \\ \downarrow && \downarrow \\ * &\stackrel{u}{\to}& U }

We think of EE as being the disjoint union over UU of the E uE_u. In the language of indexed categories, this is precisely the case: the object ESETE\in SET is the indexed coproduct of the UU-indexed family (EU)SET/U(E\to U) \in SET/U.

  • By the definition of UU-smallness and the notation just introduced, an object SS in SETSET, regarded as a **-indexed family S*S \to *, is UU-small precisely if it is isomorphic to one of the E uE_u.

  • If SS is a UU-small set by the above and if S 0SS_0 \hookrightarrow S is a monomorphism so that S 0S_0 is a subset of SS, it follows from 1) and 2) that the comoposite (S 0S*)=(S 0*)(S_0 \hookrightarrow S \to *) = (S_0 \to *) is UU-small, hence that S 0S_0 is UU-small. So: a subset of a UU-small set is UU-small.

    • In particular, let \emptyset be the initial object, which is a subset Ω\emptyset \hookrightarrow \Omega of Ω=2\Omega = \mathbf{2}. So: the empty set is UU-small.
  • Let SS, TT and KK be objects of SETSET, regarded as **-indexed families f:S*f\colon S \to *, T*T \to * and K*K \to *. Notice that (SETS)(f *K,f *T)(SETS)(K×S p 2 S,T×S p 2 S)(SET\downarrow S)(f^* K, f^* T) \simeq (SET\downarrow S)(\array{K \times S \\ \downarrow^{p_2} \\ S}, \array{T \times S \\ \downarrow^{p_2} \\ S}) is canonically isomorphic to SET(K×S,T)SET(K \times S, T). Since Π f\Pi_f is defined to be the right adjoint to f *:SETSETSf^*\colon SET \to SET \downarrow S it follows that Π ff *TT S\Pi_f f^* T \simeq T^S is the function set of functions from SS to TT. By 3), if SS, TT are UU-small then so is the function set T ST^S.

    • Since by 4) Ω=2\Omega = \mathbf{2} is UU-small and for every SS the function set 2 SP(S)\mathbf{2}^S \simeq P(S) is the power set of SS, it follows that the power set of a UU-small set is UU-small.
  • Let II be a UU-small set, in that I*I \to * is UU-small, and let SIS \to I be UU-small, to be thought of as an II-indexed family of UU-small sets S iS_i, where S iS_i is the pullback S i S * i I\array{ S_i &\to& S \\ \downarrow && \downarrow \\ * &\stackrel{i}{\to}& I }, so that SS is the disjoint union of the S iS_i: S=sqcup iIS iS = sqcup_{i \in I} S_i. By axiom 2) the composite morphism (SI*)=(S*)(S \to I \to *) = (S \to *) is UU-small, hence SS is a UU-small set, hence the II-indexed union of UU-small sets iIS i\sqcup_{i \in I} S_i is UU-small.

By standard constructions in set theory from these properties the following further closure properties of the universe UU follow.

  • For II a UU-small set and SIS \to I an II-indexed family of UU-small sets S iS_i, the cartesian product iIS i\prod_{i \in I} S_i is UU-small, as it is a subset of P(I×S)P(I \times S).

In summary

  • the sets ,*,2\emptyset, *, \mathbf{2} are UU-small;

  • a subset of a UU-small set is UU-small;

  • the power set P(S)P(S) of any UU-small set is UU-small;

  • the function set T ST^S for any two UU-small sets is UU-small;

  • the union of a UU-small family of UU-small sets is UU-small.

  • the product of a UU-small family of UU-small sets is UU-small.

Axioms of universes

Just as ZFC and other material set theories may be augmented with axioms guaranteeing the existence of Grothendieck universes, so may ETCS and other structural set theories be augmented with axioms guaranteeing the existence of universes in the above sense. For example, the counterpart of Grothendieck’s axiom

  • For every set ss there exists a universe UU containing ss, i.e. sUs\in U

would be

  • For every morphism a:AIa\colon A \to I in \mathcal{E}, there exists a universe el:EUel\colon E \to U such that aa is UU-small.

Consequences

One can show, from the above axioms, that the UU-small morphisms are closed under finite coproducts and under quotient objects. See the reference below.

In terms of indexed categories

Recall that an \mathcal{E}-indexed category is a pseudofunctor opCat\mathcal{E}^{op}\to \Cat. The fundamental \mathcal{E}-indexed category is the self-indexing 𝔼\mathbb{E} of \mathcal{E}, which takes II\in \mathcal{E} to the slice category 𝔼 I=/I\mathbb{E}^I = \mathcal{E}/I and x:IJx\colon I \to J to the base change functor x *x^*.

An internal full subcategory of \mathcal{E} is a full sub-indexed category 𝔽\mathbb{F} of 𝔼\mathbb{E} (that is, a collection of full subcategories 𝔽 I𝔼 I\mathbb{F}^I\subset \mathbb{E}^I closed under reindexing) such that there exists a generic 𝔽\mathbb{F}-morphism, i.e. a morphism el:EUel\colon E \to U in 𝔽 U\mathbb{F}^U such that for any a:AIa\colon A \to I in 𝔽 I\mathbb{F}^I, we have af *(el)a \cong f^*(el) for some f:IUf\colon I \to U. In this case (since \mathcal{E} is locally cartesian closed) there exists an internal category U 1UU_1 \;\rightrightarrows\; U in \mathcal{E} such that 𝔽\mathbb{F} is equivalent, as an indexed category, to the indexed category represented by U 1UU_1 \;\rightrightarrows\; U.

An internal full subcategory is an internal full subtopos if each 𝔽\mathbb{F} is a logical subtopos of 𝔼\mathbb{E} (closed under finite limits, exponentials, and containing the subobject classifier). A universe in \mathcal{E}, as defined above, can then be identified with an internal full subtopos satisfying the additional axiom that UU-small morphisms are closed under composition.

In the internal logic

In a topos with a universe, we can talk about small objects in the internal logic by instead talking about elements of UU. We can then rephrase the axioms of a universe in the internal logic to look more like the usual axioms for a Grothendieck universe, with the morphism el:EUel\colon E \to U interpreted as a “family of objects” (S u) u:U(S_u)_{u\colon U}:

  1. for all uu in UU, if XX is a subset of S uS_u (in the sense that there exists an injection XS uX \embedsin S_u), then there is a vv in UU such that XS vX \cong S_v;
  2. for all uu in UU, there is a vv in UU such that the power set P(S u)S vP(S_u) \cong S_v;
  3. there is a vv in UU such that S vS_v is empty;
  4. for all uu in UU and all functions f:S uUf\colon S_u \to U, there is a vv in UU such that the disjoint union i:S uS f(i)S v\biguplus_{i\colon S_u} S_{f(i)} \cong S_v.

In a well-pointed topos, such as a model of ETCS, these “internal” axioms are equivalent to their “external” versions that refer to global elements of UU.

Mike: I haven’t actually checked anything in this section, but it seems probable.

References

  • Thomas Streicher, Universes in Toposes, In: From sets and types to topology and analysis: towards practicable foundations for constructive mathematics (ps,pdf)

Revised on October 12, 2012 04:30:40 by David Roberts (192.43.227.18)